Approximation of the controls for the wave equation with a potential
Approximation of the controls for the wave equation with a potential
复制标题
具有势能的波动方程的控制逼近
DOI：
10.1007/s00211-020-01106-2
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发表时间：
2020
影响因子：
2.1
通讯作者：
Laurentiu Emanuel Temereanca
中科院分区：
文献类型：
--
作者：
S. Micu;Ionel Rovența;Laurentiu Emanuel Temereanca
本文使用有限差分空间半离散格式来处理具有变势的一维线性波动方程的边界控制的近似。由于高频数值寄生振荡，半离散模型在网格尺寸方面不是均匀可控的，并且不能保证近似控制的收敛性。在本文中，我们分析了要控制的初始数据及其离散化如何影响控制的近似。在势的某些条件下，我们证明如果预先滤除离散初始数据的最高频率，则可以确保该方案的收敛性。提出并分析了几种过滤程序。此外，我们确定了一类（常规）连续初始数据，无需任何特殊处理即可统一控制。
This article deals with the approximation of the boundary controls of a 1-D linear wave equation with a variable potential by using a finite difference space semi-discrete scheme. Due to the high frequency numerical spurious oscillations, the semi-discrete model is not uniformly controllable with respect to the mesh-size and the convergence of the approximate controls cannot be guaranteed. In this paper we analyze how do the initial data to be controlled and their discretization affect the approximation of the controls. Under certain conditions on the potential, we prove that the convergence of the scheme is ensured if the highest frequencies of the discrete initial data have been previously filtered out. Several filtration procedures are proposed and analyzed. Moreover, we identify a class of (regular) continuous initial data which can be controlled uniformly without any special treatment.